An engine pumps water continously through a hose. Water leave the hose with a velocity `v` and `m` is the mass per unit length of the Water jet. What is the rate at Which kinetic energy is imparted to water?

Options

A. (a) `1/2kv^2`
B. (b) `1/2kv^3`
C. (c) `(v^2)/(2k)`
D. (d) `(v^3)/(2k)`

Explanation

To find the rate at which kinetic energy is imparted to the water being pumped through the hose, we can follow these steps:

### Step 1: Understand the Kinetic Energy Formula
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \(m\) is the mass of the object and \(v\) is its velocity.

### Step 2: Define the Mass Flow Rate
The mass flow rate (\(\frac{dm}{dt}\)) is the mass of water that flows through the hose per unit time. Given that \(m\) is the mass per unit length of the water jet, we can express the mass flow rate as:
\[ \frac{dm}{dt} = m \cdot \frac{dl}{dt} \]
where \(\frac{dl}{dt}\) is the velocity of the water jet, which is \(v\).

### Step 3: Substitute for Mass Flow Rate
Substituting \(\frac{dl}{dt} = v\) into the mass flow rate equation gives:
\[ \frac{dm}{dt} = m \cdot v \]

### Step 4: Calculate the Rate of Kinetic Energy
Now, we can find the rate at which kinetic energy is imparted to the water. This is given by:
\[ \text{Rate of KE} = \frac{d(KE)}{dt} = \frac{1}{2} \frac{dm}{dt} v^2 \]
Substituting \(\frac{dm}{dt} = m \cdot v\) into the equation:
\[ \text{Rate of KE} = \frac{1}{2} (m \cdot v) v^2 \]

### Step 5: Simplify the Expression
Simplifying the expression gives:
\[ \text{Rate of KE} = \frac{1}{2} m v^3 \]

### Final Answer
Thus, the rate at which kinetic energy is imparted to the water is:
\[ \text{Rate of KE} = \frac{1}{2} m v^3 \]

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